The WingFuselage System in Supersonic Incident Flow
General remarks Numerous contributions to the aerodynamics of the wingfuselage interference at supersonic velocities have been published. However, the establishment of a simple, generally valid method for its computation, like the method already available for incompressible flow (Sec. 62), has not yet been devised. Summary presentations have been given by Lawrence and Flax [26], Ferrari [6], Pitts et al. [37], and Ashley and Rodden [2].
The earlier theories on the wingfuselage interferences were based on the assumptions made for the theories of the wing and of the fuselage and were limited to specific wingfuselage systems. The first work of this kind came from Kirkby and Robinson [5]. Here a wing of large aspect ratio, attached to a conical body, is treated by means of the stripe method. This method does not take into account the lift loss at the wingfuselage interface and the effect of the wing on the fuselage due to the large ratio of wing span to fuselage diameter. According to Cramer [4], these two contributions cancel each other to a large extent and the total lift is obtained relatively well. Fundamental investigations in the field of wingfuselage interference at supersonic velocities have been conducted by Ferrari [5]. He was concerned with the problem of a rectangular wing of large aspect ratio on a cylindrical fuselage with a pointed nose. The solution is accomplished through an iteration procedure. After having determined separately the potential functions for the wing and for the fuselage, these functions are combined and corrected step by step in such a way that the boundary conditions are satisfied exactly for one part of the system only, either for the wing or for the fuselage, whereas the boundary conditions of the other part are disregarded. This procedure converges after a few steps.
Browne et al. [3] investigated a delta wing with conical fuselage (wing and fuselage apex coincide) by the method of conesymmetric flows. Results are given for wings with subsonic and supersonic leading edges. Although this method does not offer extensive practical applications, its exact solutions are valuable for comparisons with approximation solutions. The results of Lock [28] illustrate this fact.
A rectangular wing with a cylindrical fuselage has been treated by Morikawa
[31] and Nielsen [37]. By applying the Laplace transform, an exact solution is obtained in the form of a series. Morikawa offers only an approximate solution, but Nielsen succeeded in re transforming the solution. The computational procedure of Woodward [52], applicable to subsonic and supersonic flows as well, should be mentioned.
Now it will be shown that the essential relationships may be gained through simple, physically plain considerations without lengthy mathematical derivations; see also Schrenk [43].
In analogy to incompressibly flow, the wingfuselage interference for supersonic flow will be analyzed by first discussing the effect of the wing on the fuselage and then the effect of the fuselage on the wing.
Lift distribution of the fuselage As has been shown in Sec. 533, the lift distribution at supersonic incident flow of the fuselage alone may be determined from the relationship for incompressible flow, by setting a(x) = aoo = const in Eq. (528), as
In Fig. 632n, the supersonic flow for the simple wingfuselage system of an axisymmetric fuselage and a rectangular wing in midwing position is demonstrated schematically. The absence of an influence of the wing on the fuselage portion before the wing in supersonic flow marks the important difference between supersonic incident flow and incompressible flow. Hence, the lift distribution for the front portion of the fuselage in a wingfuselage system is identical to that of the fuselage alone as given by Eq. (635). Thus the lift of the fuselage front portion becomes
LFf=2’noLaaqmR (636)
For the remaining portion of the fuselage, a simple survey will be given of the lift distribution created in addition to Eq. (635) by the wing on the fuselage. To simplify the problem, a wing of infinite span has been assumed in Fig. 632. In this case the wing generates perturbation velocities only in the range between the Mach lines mі and m2 originating at its leading edge and its trailing edge. Under the simplifying assumption that the lift distribution of the wing is unchanged in the fuselage range, no additive lift force, caused by the wing, acts on the rear fuselage portion either. Hence the fuselage feels an additive lift force only within the range between the Mach lines тг and rn2.
This lift due to the wing influence is caused by the velocities induced by the wing in the x direction, that is, u(x), and in the z direction, that is, w(x). In Fig. 632Z? and c, the distribution of the induced velocities u(x) and w(x) is given. Their variation on the fuselage surface at z = 0 and # = 90°, respectively, is marked by the dashed curve, and that at у = 0 and $=0°, respectively, by the dashdotted curve. These two curves are merely displaced from each other in the longitudinal
Figure 632 Computation of the interference of wingfuselage systems at supersonic incident flow, (a) Geometry of the wingfuselage system. (b) Distribution of the longitudinal velocity u(x). (c) Distribution of the vertical velocity w(jc). (d) Lift distribution due to the longitudinal velocity, (e) Lift distribution due to the vertical velocity. 00 Resultant lift distribution.
direction. The maximum values of the induced longitudinal and vertical velocities are given in Eqs. (441tz) and (441 b) as [31]
fuselage. The following expressions from Ferrari [5] are obtained for 0<x <x0, if in this range R(x) =R0 = const:
Here x0 =R0 ^/MaL — 1. Corresponding formulas are valid for с <x < c +• x0.
From these mean values of the induced velocities, two contributions are obtained to the lift distribution of the fuselage in the wing range, namely, (dLp/dx)і from the longitudinal velocity й(х), and {dbpldx)2 from the vertical velocity OieoUoo + w(x). For the case of a constant fuselage radius R(x) —R0 within the wing range 0 <x <c 4 xQ, these contributions are
Ш,**•!? <«■>
(^ = 2m™R2°&c
The qualitative trend of these two contributions is shown in Fig. 632d and e. The resulting lift distribution as the sum of these two contributions is presented in Fig. 632/. Integration of the contribution of Eq. (6396) results in LF2 = 0, because, according to Fig. 632c, w(x) is equal to zero before and behind the wing. From Fig. 6326, the lift force is obtained through integration of Eq. (639a), because Lpi = Lp, as
This simple relationship for the lift of the fuselage due to the wing applies to the cases where the intersection of the front Mach line mi with the fuselage surface at у = 0 lies before the wing trailing edge (see Fig. 632a). There is another interpretation of this result of Eq. (640), namely, that the fuselage lift due to wing effects is equal to the lift in plane flow of the wing portion A A = 2R0c shrouded by the fuselage. Also, the case has been investigated by Ferrari [6] where the front Mach line intersects the fuselage upper edge behind the wing trailing edge. In this case the computation of the additive fuselage lift is considerably more difficult than explained above.
In Fig. 633 the lift distribution of the fuselage under the influence of the wing is shown for an example of Cramer [4]. The theoretical curve has been computed by Ferrari [5]. At the Mach number Ma^ = 2 used in this study and the chosen geometry of the wingfuselage system, the Mach line from the leading edge of the wing – intersects the upper edge of the fuselage behind the wing trailing edge. Therefore, contrary to Fig. 632f, the lift distribution reaches far beyond the wing trailing edge. Agreement between theory and measurement is good.
Figure 633 Lift distribution on the fuselage for a wingfuselage system (midwing airplane) at supersonic velocities, from Cramer. Mach number = 2, angle of attack a = 8°. Theory from Ferrari, wing chord c = 1.4/?0, wing span b = 8Ra. 
As stated earlier, the above theoretical results apply to the case of wings of very large span. The effect of the wing aspect ratio can be seen in Fig. 634, where, from [6], the additive lift distributions of the wing are plotted for wingfuselage systems with wings of several aspect ratios. The Mach number is also Max = 2. When the aspect ratio decreases, the additive lift force decreases considerably, as would be expected.
For the test series of Fig. 634, the ratio of the fuselage lift Lp and the lift of
Figure 634 Effect of wing aspect ratio on the lift distribution on the fuselage for wingfuselage systems (midwing airplane) at supersonic velocities, from Ferraxi. Mach number Ma0a = 2, angle of attack сїоо = 8°. 
Figure 635 Ratio of fuselage lift Lpto wing lift L’yj vs. relative fuselage width rip for wingfuselage systems at supersonic velocities, from Ferrari (system as in Fig. 634). Mach number Ma«, = 2, angle of attack a» = 8°. Curve 1, theory from Lennertz. Curve 2, slenderbody theory.
the wing portion not shrouded by the fuselage, JL’w, is plotted in Fig. 635. The data are compared with theoretical curves: Curve 1 reflects the theory of Lennertz [27] from Eq. (65), valid also for supersonic flow. Curve 2 gives the theory of wingfuselage systems with wings of small aspect ratio of Sec. 64. The two theoretical curves are not very different. The measured data agree quite well with the theoretical curve (2).
The lift distribution leads to the pitching moment from Eq. (532). By introducing the two contributions of the lift distribution from Eqs. (639u) and (639b), the following two contributions to the pitching moment, referred to the middle of the wing, are obtained, where the contribution of the fuselage front portion has been disregarded:
Mfi = – 8 f) dx 0 
(641 a) 
(64 Щ 

Cr! Ca 0 
(642 a) 
— — Ялсі^а^Щс 
(642b) 
Here a: is measured from the wing leading edge. From Fig. 632e, Mp2 is a free moment. The total moment (without ihe fuselage front portion), referred to the middle of the wing, thus becomes
Mp = —4 ті cj.^ aoc RqC (643)
Note that this additive moment is independent of the Mach number.
The neutralpoint displacement due to the wing influence on the fuselage with reference to the neutral point of the wing alone, xNW — cl2, becomes
O^JVOCW’ + F) – “ і
b(W+F)
For small relative fuselage widths r)F = 2R0/b, L(w+f) may be approximated by Lw. For the wing of large aspect ratio, from Eq. (446), this leads to
(4xn)(w+f) n 2 a rm.—————– 7
———————— ———————— = — rjpAyMaio – 1
This stabilizing neutralpoint displacement due to the effect of the wing on the fuselage counteracts the destabilizing contribution of the fuselage front portion.
The above results on the effect of the wing are valid for the unswept wing of large aspect ratio, that is, for wings with supersonic leading edges. For wings with small aspect ratio, the discussions of Sec. 64 should be examined.
4ocqq ^ _L_ Aocjy) – 1 ‘ [32]0O 
Lift distribution of the wing The effect of the fuselage on the lift distribution of the wing at supersonic velocities can be determined approximately by the method applied in Sec. 622 to incompressible flow. The additive angleofattack distribution, caused by the cross flow over the fuselage as in Fig. 65b, creates additive lift locally on the wing. Under the assumption of an infinitely long fuselage, the additive angleofattack distribution for a given fuselage crosssection shape is the same as in incompressible flow, because the velocity of the cross flow of the fuselage is considerably lower than the speed of sound. Equations (616a) and (616b) give the distribution of the induced angle of attack for a fuselage of circular cross section (radius R) with a wing in midwing position. The computation of the approximate lift distribution along the span for the given angleofattack distribution may be conducted very easily with the socalled stripe method.* Hence, the local lift coefficient becomes
with Aa(y) from Eq. (616a).
An example for a wingfuselage system of an axisymmetric fuselage and a rectangular wing is given in Fig. 636. It shows the lift distribution plotted against the span for the Mach number Маж = 2 at an angle of attack a„ = 8°. Curve 1 reflects the theory of the stripe method, Eq. (645), and curve 2 a theory of Ferrari [6]. Both theories agree quite well with the measurements, except for the stripe method in the immediate vicinity of the fuselage. For comparison, the theory for
Figure 636 Lift distribution on the wing due to the fuselage effect for a wingfuselage system (midwing airplane) at supersonic velocities. Curve 1, theory, stripe method, from Eq. (645). Curve 2, theory from Ferrari. Curve 3, measurements from Ferrari. Curve 4, theory, wing alone.
the wing alone is added as curve 4. Obviously, the influence of the fuselage on the lift distribution of the wing is rather large.
The above results on the effect of the fuselage on the lift distribution of the wing apply to wings of large aspect ratios. For wings of small aspect ratios, reference should again be made to Sec. 64.
Wave drag The problem of the determination of the wave drag of wingfuselage systems at supersonic velocities has been attacked by Vandrey [48], Lomax and Heaslet [30], Jones [18], and Keune and Schmidt [19]. Also, the experimental investigations of Schneider [42] should be mentioned.
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